The centre of mass is also known as the centre of gravity.
The centre of mass of a body has the property that the total moment of the object’s weight about any line through the centre of mass is zero.
If a body is freely suspended from a point on the body and hangs in stable equilibrium, it will hang so that the centre of mass is directly below the point of attachment.
In this sketch, a body is suspended from the point \(A\), and its centre of mass \(G\) lies directly below \(A\).
For a planar shape (a lamina), the centre of mass is also the point on which it could be balanced horizontally.
If masses are given with a coordinate system, say there is a mass of \(m_1\) at \((x_1,y_1)\), a mass of \(m_2\) at \((x_2,y_2)\), …, and a mass of \(m_n\) at \((x_n,y_n)\), then the moments property above can be used to calculate the centre of mass of the whole collection. It turns out that this centre of mass, \((\bar x,\bar y)\), has the property that \(\bar x\) is the weighted mean of the \(x_i\) (weighted by the masses \(m_i\)), and similarly for \(\bar y\). So \[\bar x=\frac{\sum_{i=1}^n m_ix_i}{\sum_{i=1}^n m_i}\qquad\text{and}\qquad \bar y=\frac{\sum_{i=1}^n m_iy_i}{\sum_{i=1}^n m_i}.\]
